Finite Group Databasebeta

← C3×#25856 #10644463

Subgroups — C3×#25856 #10644463

Order 768.

1338 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic
2C2×C3×#487C212yesmaximal, complemented
3C3×#2857C211yesmaximal, characteristic
4C3×#2043C214yesmaximal, complemented
5C3×#1956C212yesmaximal, complemented
6C3×#1787C214yesmaximal
7C3×C4×C22⋊C8C211yesmaximal, characteristic
8#25856C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×Q8C2212yes—
10C2×C3×C4⋊Q8—28no—
11C2×C3×C4⋊Q8—21no—
12C2×C3×C4⋊Q8—21no—
13C2×C3×C4⋊Q8C2212yes—
14C2×C3×C4⋊Q8C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C22⋊Q8—21no—
17C22×C3×C4⋊C4—21no—
18C22×C3×C4⋊C4—21no—
19C22×C3×C42C2211yescharacteristic
20C3×#487—28no—
21C2×C3×#103C2218yes—
22C2×C3×#103C2218yes—
23C2×C3×C22⋊C8C2212yes—
24C2×C3×C4×C8C2212yes—
25C3×#391—21no—
26C3×#389—21no—
27C3×#389C2212yes—
28C3×#384C2211yescharacteristic
29C3×#380—21no—
30C3×#380C2212yes—
31C3×#336C2212yes—
32C22×C4⋊Q8C2×C311yescharacteristic
33C2×#487C2×C312yescomplemented
34#2857C2×C311yescharacteristic
35#2043C2×C314yescomplemented
36#1956C2×C312yescomplemented
37#1787C2×C314yes—
38C4×C22⋊C8C2×C311yescharacteristic
39C22×C3×Q8—28no—
40C22×C3×Q8—22no—
41C22×C3×Q8—21no—
42C22×C3×Q8—28no—
43C22×C3×Q8—44no—
44C22×C3×Q8C2314yes—
45C23×C3×C4—22no—
46C23×C3×C4D411yescharacteristic
47C23×C3×C4D411yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C22×C3×C8C2314yes—
50C3×C4⋊Q8—28no—
51C3×C4⋊Q8—28no—
52C3×C4⋊Q8—28no—
53C3×C4⋊Q8—24no—
54C3×C4⋊Q8—24no—
55C3×C4⋊Q8D418yes—
56C3×C22⋊Q8—44no—
57C2×C3×C4⋊C4—28no—
58C2×C3×C4⋊C4—28no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C22⋊C4—22no—
70C2×C3×C42—24no—
71C2×C3×C42D411yescharacteristic
72C2×C3×C42D411yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C3×#103—216no—
76C3×#103—48no—
77C3×C22⋊C8—24no—
78C3×C4×C8—24no—
79C3×#95—22no—
80C3×#95—22no—
81C23×Q8C22×C312yes—
82C2×C4⋊Q8—28no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8C22×C312yes—
86C2×C4⋊Q8C22×C312yes—
87C2×C4⋊Q8C22×C311yescharacteristic
88C2×C22⋊Q8—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C42C22×C311yescharacteristic
92#487—28no—
93C2×#103C22×C318yes—
94C2×#103C22×C318yes—
95C2×C22⋊C8C22×C312yes—
96C2×C4×C8C22×C312yes—
97#391—21no—
98#389—21no—
99#389C22×C312yes—
100#384C22×C311yescharacteristic
101#380—21no—
102#380C22×C312yes—
103#336C22×C312yes—
104C24×C3C2×D411yescharacteristic
105C2×C3×Q8—28no—
106C2×C3×Q8—24no—
107C2×C3×Q8—28no—
108C2×C3×Q8—28no—
109C2×C3×Q8—432no—
110C2×C3×Q8—44no—
111C2×C3×Q8—42no—
112C2×C3×Q8—44no—
113C2×C3×Q8—42no—
114C2×C3×Q8—42no—
115C2×C3×Q8C2×D418yes—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—28no—
123C22×C3×C4—24no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—44no—
127C22×C3×C4—44no—
128C22×C3×C4SD1614yes—
129C22×C3×C4Q1614yes—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4#4114yes—
136C22×C3×C4C2411yescharacteristic
137C2×C3×C8—24no—
138C2×C3×C8—24no—
139C2×C3×C8—24no—
140C3×C4⋊C4—28no—
141C3×C4⋊C4—28no—
142C3×C4⋊C4—28no—
143C3×C4⋊C4—28no—
144C3×C4⋊C4—24no—
145C3×C4⋊C4—24no—
146C3×C4⋊C4—42no—
147C3×C4⋊C4—42no—
148C3×C4⋊C4—44no—
149C3×C4⋊C4—42no—
150C3×C22⋊C4—44no—
151C3×C42—22no—
152C3×C42—22no—
153C3×C42—22no—
154C3×C42C2×D414yes—
155C22×Q8—28no—
156C22×Q8—22no—
157C22×Q8—21no—
158C22×Q8—28no—
159C22×Q8—44no—
160C22×Q8C23×C314yes—
161C23×C4—22no—
162C23×C4C3×D411yescharacteristic
163C23×C4C3×D411yescharacteristic
164C23×C4C23×C311yescharacteristic
165C22×C8C23×C314yes—
166C4⋊Q8—28no—
167C4⋊Q8—28no—
168C4⋊Q8—28no—
169C4⋊Q8—24no—
170C4⋊Q8—24no—
171C4⋊Q8C3×D418yes—
172C22⋊Q8—44no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C23×C312yes—
184C2×C4⋊C4C23×C312yes—
185C2×C22⋊C4—22no—
186C2×C42—24no—
187C2×C42C3×D411yescharacteristic
188C2×C42C3×D411yescharacteristic
189C2×C42C23×C311yescharacteristic
190C2×C42C23×C311yescharacteristic
191#103—216no—
192#103—48no—
193C22⋊C8—24no—
194C4×C8—24no—
195#95—22no—
196#95—22no—
197C23×C3—21no—
198C23×C3—21no—
199C23×C3—21no—
200C23×C3—21no—
201C23×C3#12412yes—
202C23×C3C2×SD1612yes—
203C23×C3C2×Q1612yes—
204C23×C3C22×D411yescharacteristic
205C3×Q8—216no—
206C3×Q8—416no—
207C3×Q8—48no—
208C3×Q8—84no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—28no—
216C2×C3×C4—28no—
217C2×C3×C4—28no—
218C2×C3×C4—28no—
219C2×C3×C4—416no—
220C2×C3×C4—44no—
221C2×C3×C4—44no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4#12012yes—
229C2×C3×C4#12012yes—
230C2×C3×C4#12412yes—
231C2×C3×C4C2×SD1612yes—
232C2×C3×C4C2×SD1612yes—
233C2×C3×C4C2×Q1612yes—
234C2×C3×C4C2×Q1612yes—
235C2×C3×C4#13712yes—
236C2×C3×C4#13712yes—
237C2×C3×C4C22×D412yes—
238C2×C3×C4C2×#4112yes—
239C2×C3×C4H(5,𝔽2)12yes—
240C3×C8—44no—
241C24C2×C3×D411yescharacteristic
242C2×Q8—28no—
243C2×Q8—24no—
244C2×Q8—28no—
245C2×Q8—28no—
246C2×Q8—432no—
247C2×Q8—44no—
248C2×Q8—42no—
249C2×Q8—44no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8C2×C3×D418yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4C3×SD1614yes—
266C22×C4C3×Q1614yes—
267C22×C4C2×C3×D411yescharacteristic
268C22×C4C2×C3×D411yescharacteristic
269C22×C4C2×C3×D411yescharacteristic
270C22×C4C2×C3×D411yescharacteristic
271C22×C4C2×C3×D411yescharacteristic
272C22×C4C3×#4114yes—
273C22×C4C24×C311yescharacteristic
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C4⋊C4—28no—
280C4⋊C4—28no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—42no—
287C22⋊C4—44no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42C2×C3×D414yes—
292C22×C3—22no—
293C22×C3—22no—
294C22×C3—22no—
295C22×C3—22no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3#48714yescomplemented
299C22×C3C2×#12011yescharacteristic
300C22×C3C2×#12411yescharacteristic
301C22×C3#53511yescharacteristic
302C22×C3C22×SD1611yescharacteristic
303C22×C3C22×Q1611yescharacteristic
304C22×C3C2×#13711yescharacteristic
305C22×C3C2×#13711yescharacteristic
306C3×C4—22no—
307C3×C4—22no—
308C3×C4—22no—
309C3×C4—48no—
310C3×C4—82no—
311C3×C4—82no—
312C3×C4#44814yes—
313C3×C4#45114yes—
314C3×C4#54614yes—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23C3×#12412yes—
320C23C2×C3×SD1612yes—
321C23C2×C3×Q1612yes—
322C23C22×C3×D411yescharacteristic
323Q8—216no—
324Q8—416no—
325Q8—48no—
326Q8—84no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—28no—
334C2×C4—28no—
335C2×C4—28no—
336C2×C4—28no—
337C2×C4—416no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4C3×#12012yes—
347C2×C4C3×#12012yes—
348C2×C4C3×#12412yes—
349C2×C4C2×C3×SD1612yes—
350C2×C4C2×C3×SD1612yes—
351C2×C4C2×C3×Q1612yes—
352C2×C4C2×C3×Q1612yes—
353C2×C4C3×#13712yes—
354C2×C4C3×#13712yes—
355C2×C4C22×C3×D412yes—
356C2×C4C2×C3×#4112yes—
357C2×C4C3×H(5,𝔽2)12yes—
358C8—44no—
359C2×C3—24no—
360C2×C3#243511yescharacteristic
361C2×C3C2×#44811yescharacteristic
362C2×C3C2×#45111yescharacteristic
363C2×C3C2×#48711yescharacteristic
364C2×C3#313911yescharacteristic
365C2×C3#320611yescharacteristic
366C2×C3#320911yescharacteristic
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22C3×#48714yescomplemented
374C22C2×C3×#12011yescharacteristic
375C22C2×C3×#12411yescharacteristic
376C22C3×#53511yescharacteristic
377C22C22×C3×SD1611yescharacteristic
378C22C22×C3×Q1611yescharacteristic
379C22C2×C3×#13711yescharacteristic
380C22C2×C3×#13711yescharacteristic
381C4—22no—
382C4—22no—
383C4—22no—
384C4—48no—
385C4—82no—
386C4—82no—
387C4C3×#44814yes—
388C4C3×#45114yes—
389C4C3×#54614yes—
390C3#2585611yescharacteristic, direct product factor, Sylow
391C2—24no—
392C2C3×#243511yescharacteristic
393C2C2×C3×#44811yescharacteristic
394C2C2×C3×#45111yescharacteristic
395C2C2×C3×#48711yescharacteristic
396C2C3×#313911yescharacteristic
397C2C3×#320611yescharacteristic
398C2C3×#320911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×#487C2semidirect
4. C3×#2043C2semidirect
5. C3×#1956C2semidirect
8. #25856C3direct (N × H)
33. C2×#487C2×C3semidirect
35. #2043C2×C3semidirect
36. #1956C2×C3semidirect
298. C22×C3#487semidirect
373. C22C3×#487semidirect
390. C3#25856direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C3×C4⋊Q82
C2×C3×#4872
C3×#28572
C3×#20432
C3×#19562
C3×#17872
C3×C4×C22⋊C82
#258563